Eigenvectors of hermitian matrix are not coming out to be orthogonal

Question

I have the following hermitian matrix before me: \begin{pmatrix}2 &1+i\\ 1-i& 3\end{pmatrix} I calculated its characteristic polynomial as $k^2-5k+4$ which has $1$ and $4$ as its roots. Eigenvectors corresponding to these two eigenvalues are \begin{pmatrix}-(1+i) &1 \end{pmatrix} and \begin{pmatrix}1 &1-i\end{pmatrix} But these are not orthogonal. Why is it so? Am I doing something wrong?

Answer 1

Check Eigen values are $4,1$ and eigenvectors are $V_1=\begin{bmatrix} 1+i \\2 \end{bmatrix}$ and $V_2=\begin{bmatrix} -1-i \\ 1 \end{bmatrix}$, then $V_1^{\dagger} V_2=0$, $\dagger$ means conjugate and transpose.